29 lines
2.6 KiB
Markdown
29 lines
2.6 KiB
Markdown
- Radian:
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- **Def**: *It is the SI unit for measuring angles (in the plane).*
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- **Def**: *$1$ radian is defined as the angle subtended at the center by a circular arc of length equal to the radius*
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- **Def**: *A general angle is measured in radians as the ration of the length an associated circular arc and the corresponding radius. That is $\theta=\frac{s}{r}\text{rad}$*
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- **Def**: *Usually "$rad$" is omitted.*
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- Ex: $$\begin{align}180^\circ=\pi\text{ rad}\\\frac{\pi}{3}\text{ rad}=30^\circ\\\frac{\pi}{4}\text{ rad}=45^\circ\\\frac{\pi}{3}\text{ rad}=60^\circ\\\frac{\pi}{2}\text{ rad}=90^\circ\\2\pi\text{ rad}=360^\circ\end{align}$$
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- The right angled triangle
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- **Def**: *The trigonometric functions: *$$\begin{align}\sin\theta=\frac{\text{perpendicular}}{\text{hypotenuse}}\\\cos\theta=\frac{\text{base}}{\text{hypotenuse}}\\\tan\theta=\frac{\text{perpendicular}}{\text{base}}\end{align}$$
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- In addition to above, $\csc\theta=\frac{1}{\sin\theta},\sec\theta=\frac{1}{\cos\theta},\cot\theta=\frac{1}{\tan\theta}$
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- Pythagoras' formula: $p^2+b^2=h^2$
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which leads to the **trigonometric identity**: $\sin^2\theta+\cos^2\theta=1$
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and also $\tan^2\theta+1=\sec^2\theta$
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- Dominains and ranges:
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- $D_{\sin}=\mathbb{R}\;\;R_{\sin}=[-1,1]$
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- $D_{\cos}=\mathbb{R}\;\;R_{\cos}=[-1,1]$
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- $D_{\tan}=\mathbb{R}\setminus\{n\pi+\frac{\pi}{2}:n\in\mathbb{Z}\}\;\;R_{\tan}=(-\infty,\infty)$
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- Useful relations
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- $\sin(-\theta)=-\sin(\text{odd}),\cos(-\theta)=\cos\theta(\text{even})$
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- Periodicity: $\sin(\theta+2n\pi)=\sin\theta,\cos(\theta+2n\pi)=\cos\theta,\tan(\theta+n\pi)=\tan\theta$
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- Complementary angles: $\sin(\frac{\pi}{2}-\theta)=\cos\theta,\cos(\frac{\pi}{2}-\theta)=\sin\theta$
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- Sift by $\pi$: $\sin(\theta\pm\pi)=-\sin\theta,\cos(\theta\pm\pi=-\cos\theta$
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- Sum of angles: $\sin(\theta+\phi)=\sin\theta\times\cos\phi+\cos\theta\times\sin\phi,\cos(\theta+\phi)=\cos\theta\times\cos\phi-\sin\theta\times\sin\phi,\tan(\theta+\phi)=\frac{\tan\theta+\tan\phi}{1-\tan\theta\times\tan\phi}$
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- Double angle: $\sin(2\theta)=2\sin\theta\cos\theta,\tan2\theta=\frac{2\tan\theta}{1-\tan^2\theta},\cos(2\theta)=\cos^2-\sin^2\theta=2\cos^2\theta-1=1-2\sin^2\theta$
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- Half angle: $2\sin^2\frac{\theta}{2}=1-\cos\theta,2\cos^2\frac{\theta}{2}=1+\cos\theta$
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- Solving trigonometric equations
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- $\sin\theta=\sin{a}\Leftrightarrow\theta=\left\{\begin{align}a+2n\pi,n\in\mathbb{Z}\\\pi-a+2n\pi,n\in\mathbb{Z}\end{align}\right.$
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- $\cos\theta=\cos{a}\Leftrightarrow\theta=\left\{\begin{align}a+2n\pi,n\in\mathbb{Z}\\-a+2n\pi,n\in{Z}\end{align}\right.$
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- $\tan\theta=\tan{a}\Leftrightarrow\theta=a+n\pi,n\in\mathbb{Z}$
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- Ex: Solve $\sin(x+\frac{\pi}{6})=\frac{\sqrt{3}}{2}$ |